## Chemical Reaction Rates

A *rate* is a measure of how some property varies with time. Speed is a familiar rate that expresses the distance traveled by an object in a given amount of time. Wage is a rate that represents the amount of money earned by a person working for a given amount of time. Likewise, the rate of a chemical reaction is a measure of how much reactant is consumed, or how much product is produced, by the reaction in a given amount of time.

The **rate of reaction** is the change in the amount of a reactant or product per unit time. Reaction rates are therefore determined by measuring the time dependence of some property that can be related to reactant or product amounts. Rates of reactions that consume or produce gaseous substances, for example, are conveniently determined by measuring changes in volume or pressure. For reactions involving one or more colored substances, rates may be monitored via measurements of light absorption. For reactions involving aqueous electrolytes, rates may be measured via changes in a solution’s conductivity.

For reactants and products in solution, their relative amounts (concentrations) are conveniently used for purposes of expressing reaction rates. If we measure the concentration of hydrogen peroxide, H_{2}O_{2}, in an aqueous solution, we find that it changes slowly over time as the H_{2}O_{2} decomposes, according to the equation:

\({\text{2H}}_{2}{\text{O}}_{2}\left(aq\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{2H}}_{2}\text{O}\left(l\right)+{\text{O}}_{2}\left(g\right)\)

The rate at which the hydrogen peroxide decomposes can be expressed in terms of the rate of change of its concentration, as shown here:

\(\begin{array}{cc}\hfill \text{rate of decomposition of}\phantom{\rule{0.2em}{0ex}}{\text{H}}_{2}{\text{O}}_{2}& =-\phantom{\rule{0.2em}{0ex}}\cfrac{\text{change in concentration of reactant}}{\text{time interval}}\hfill \\ & =-\phantom{\rule{0.2em}{0ex}}\cfrac{{\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{2}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{1}}}{{t}_{2}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{t}_{1}}\hfill \\ & =-\phantom{\rule{0.2em}{0ex}}\cfrac{\text{Δ}\left[{\text{H}}_{2}{\text{O}}_{2}\right]}{\text{Δ}t}\hfill \end{array}\)

This mathematical representation of the change in species concentration over time is the **rate expression** for the reaction. The brackets indicate molar concentrations, and the symbol delta (Δ) indicates “change in.” Thus, \({\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{1}}\) represents the molar concentration of hydrogen peroxide at some time *t*_{1}; likewise,\({\left[{\text{H}}_{2}{\text{O}}_{2}\right]}_{{t}_{2}}\) represents the molar concentration of hydrogen peroxide at a later time *t*_{2}; and Δ[H_{2}O_{2}] represents the change in molar concentration of hydrogen peroxide during the time interval Δ*t* (that is, *t*_{2} − *t*_{1}).

Since the reactant concentration decreases as the reaction proceeds, Δ[H_{2}O_{2}] is a negative quantity; we place a negative sign in front of the expression because reaction rates are, by convention, positive quantities. The figure below provides an example of data collected during the decomposition of H_{2}O_{2}.

To obtain the tabulated results for this decomposition, the concentration of hydrogen peroxide was measured every 6 hours over the course of a day at a constant temperature of 40 °C. Reaction rates were computed for each time interval by dividing the change in concentration by the corresponding time increment, as shown here for the first 6-hour period:

\(\cfrac{-\text{Δ}\left[{\text{H}}_{2}{\text{O}}_{2}\right]}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\cfrac{-\left(\text{0.500 mol/L}-\text{1.000 mol/L}\right)}{\left(\text{6.00 h}-\text{0.00 h}\right)}\phantom{\rule{0.1em}{0ex}}=0.0833 mol\phantom{\rule{0.2em}{0ex}}{\text{L}}^{-1}\phantom{\rule{0.2em}{0ex}}{\text{h}}^{-1}\)

Notice that the reaction rates vary with time, decreasing as the reaction proceeds. Results for the last 6-hour period yield a reaction rate of:

\(\cfrac{-\text{Δ}\left[{\text{H}}_{2}{\text{O}}_{2}\right]}{\text{Δ}t}\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}\cfrac{-\left(0.0625\phantom{\rule{0.2em}{0ex}}\text{mol/L}-0.125\phantom{\rule{0.2em}{0ex}}\text{mol/L}\right)}{\left(24.00\phantom{\rule{0.2em}{0ex}}\text{h}-18.00\phantom{\rule{0.2em}{0ex}}\text{h}\right)}\phantom{\rule{0.1em}{0ex}}=0.0104\phantom{\rule{0.2em}{0ex}}\text{mol}\phantom{\rule{0.2em}{0ex}}{\text{L}}^{-1}\phantom{\rule{0.2em}{0ex}}{\text{h}}^{-1}\)

This behavior indicates the reaction continually slows with time. Using the concentrations at the beginning and end of a time period over which the reaction rate is changing results in the calculation of an **average rate** for the reaction over this time interval. At any specific time, the rate at which a reaction is proceeding is known as its **instantaneous rate**. The instantaneous rate of a reaction at “time zero,” when the reaction commences, is its **initial rate**.

Consider the analogy of a car slowing down as it approaches a stop sign. The vehicle’s initial rate—analogous to the beginning of a chemical reaction—would be the speedometer reading at the moment the driver begins pressing the brakes (*t*_{0}). A few moments later, the instantaneous rate at a specific moment—call it *t*_{1}—would be somewhat slower, as indicated by the speedometer reading at that point in time.

As time passes, the instantaneous rate will continue to fall until it reaches zero, when the car (or reaction) stops. Unlike instantaneous speed, the car’s average speed is not indicated by the speedometer; but it can be calculated as the ratio of the distance traveled to the time required to bring the vehicle to a complete stop (Δ*t*). Like the decelerating car, the average rate of a chemical reaction will fall somewhere between its initial and final rates.

The instantaneous rate of a reaction may be determined one of two ways. If experimental conditions permit the measurement of concentration changes over very short time intervals, then average rates computed as described earlier provide reasonably good approximations of instantaneous rates. Alternatively, a graphical procedure may be used that, in effect, yields the results that would be obtained if short time interval measurements were possible. If we plot the concentration of hydrogen peroxide against time, the instantaneous rate of decomposition of H_{2}O_{2} at any time *t* is given by the slope of a straight line that is tangent to the curve at that time (see the figure below). We can use calculus to evaluating the slopes of such tangent lines, but the procedure for doing so is beyond the scope of this tutorial.